The hyphenated name gives the boy one foot in the old world and one in the new – perfect for the transatlantic couple, who “decided on a name that’s great in both the U.S. and France,” a source close to Martinez recently told PEOPLE.

Berry, who won the Oscar in 2002 for Monster’s Ball, also has a 5½-year-old daughter, Nahla Ariela, from her previous relationship with Gabriel Aubry.